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Lie Elements and Knuth Relations
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- Journal:
- Canadian Journal of Mathematics / Volume 56 / Issue 4 / 01 August 2004
- Published online by Cambridge University Press:
- 20 November 2018, pp. 871-882
- Print publication:
- 01 August 2004
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- Article
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A coplactic class in the symmetric group
${{\mathcal{S}}_{n}}$ consists of all permutations in ${{\mathcal{S}}_{n}}$ with a given Schensted 
$Q$-symbol, and may be described in terms of local relations introduced by Knuth. Any Lie element in the group algebra of ${{\mathcal{S}}_{n}}$ which is constant on coplactic classes is already constant on descent classes. As a consequence, the intersection of the Lie convolution algebra introduced by Patras and Reutenauer and the coplactic algebra introduced by Poirier and Reutenauer is the direct sum of all Solomon descent algebras.
